Welcome to calculus.
I'm professor Ghrist. We're about to begin lecture 50 on
infinite series. The next few lessons are at the heart of
this chapter and deal with infinite series, a hopefully
not too unfamiliar topic. We're going to detail specific examples,
and consider convergence and divergence of series, in
the same manner that we dealt with convergence and
divergence of improper integrals. We continue with our program of building a
calculus for sequences. Considering, in this lesson, the analog of
the improper integral. These are called infinite series.
Something that you've seen before. But now, we're going to treat them a bit
more carefully. Now, how was it that we handled improper
integrals? Recall, the integral from 1 to infinity of
f of x dx was defined in terms of a limit.
The limit as t goes to infinity of the finite integral of f from 1 to t.
So, in defining what we mean by an infinite series, that is, the sum as n
goes from 1 to infinity of a sub n. We'll use the same approach. We can think of this infinite series of
being something like a discretization of an improper integral.
And so, we can define this sum to be the limit as t goes
to infinity of the sum n goes from 1 to t of a sub n. That is, the series is really the limit of
the sequence of partial sums. Recall that, when it comes to improper integrals, the central and subtle question is that of convergence or
divergence. The same occurs with infinite series. Now, you've seen infinite series all throughout this course, from the very
beginning. And I have told you repeatedly, don't
worry too much about convergence or divergence, we'll worry about that at
the end of the course. Well, it is time to push that button.
It is time for all of those worries that have been inside your head to
come out, and we will deal with them. We will go through the notion of convergence or divergence, slowly and
carefully. Giving you lots of practice, so that you
get good at it. Let's consider some examples of series.
Some are very easy to understand. If we look at 1 plus 1 over e plus 1 over
e squared plus 1 over e cubed, et cetera. Well, we recognize this as a geometric
series. So, not only do we know that it converges,
we know exactly to what value. On the other hand, we recall that the
geometric series does not always converge. For example, if we evaluate the geometric series at
negative 1, then, one could try to argue that the
appropriate value is 0, or, one could argue that the
appropriate convergence is to 1, or even to 1 half.
None of these holds.
This is a divergent series, because the limit of partial sums does not exist.
It oscillates. Now, subseries do converge, but are so
subtle in how they do so. They're very difficult to evaluate.
The sum from 1 to infinity of 1 over n squared does converge
as we'll be able to show soon. But it convereges to a value that is very difficult to
determine exactly.
And that value is pi squared over 6. Some series are not so easy to figure out.
Consider 1 plus 1 half plus 1 3rd plus 1 4th plus 1 5th, et cetera.
To what, if anything, does this converge. This last example is a special series that
is called the harmonic series. Let's investigate this series a little
bit. We'll fire up the computer and see what
happens when we add, let's say the first 10,000
terms together. We get a result that seems decidedly
finite. And it's about 9.79.
But what happens if we say add another 5,000 terms to that.
Well, our result is only a little bit bigger.
We would guess that this is about to converge,
since the next term in the series is 1 over 15,001. That is quite small.
But let's just type it. Let's add the first 20,000 terms together. Well, we can see that we're slowing down a
bit. We've only gone about 0.3 more, not even
that. Still I'm not going to be quite
comfortable until I see that it's converged to several decimal places. But this does not appear to be happening
even with 25,000 terms. So let's sum up the first 100,000 thousand
terms. Surely, surely, if this diverges and goes
off to infinity, then we would see it. Well, the answer in this case is just a
little bit more than we had before, but not so little as to make me conclude
that it converges. So what is happening in this case? Well, we're going to have to use something
beyond a calculator to determine convergence or
divergence in this case. What we're going to use is our calculus
intuition. Harmonic series reminds us of the improper integral, as x goes
from 1 to infinity of 1 over x dx.
Indeed we can consider that sum at series as a discretization of this integral. As a left Riemann sum of this integral
with a uniform partition of step size one. Now, we know that the integral of 1 over x
as x goes from 1 to infinity diverges. And we can conclude from the geometry of
this diagram if nothing else, that the harmonic series is strictly bigger.
Therefore, we can conclude that harmonic series
diverges. What's wonderful about this is that we
viewed our continuous or smooth calculus understanding to conclude
things about this discreet calculus.
Example, and this discussion raise the question, what can you trust when it
comes to determining convergence or divergence?
As we've seen, a calculator is not the best tool for determining convergence
or divergence. What about your intuition? Maybe you should just get a feel for what
works. Well, intuition is also flawed.
Let's look at an example. I claim that the series 1 minus a half
plus a third minus a fourth, plus a fifth, et
cetera, something that we'll call the alternating
harmonic series, does converge, and it converges to a value of log of 2.
Now, why is that the case? Well, recall that log of 1 plus x is
exactly this kind of alternating sum of x to the n over n.
If we evaluate this at x equals 1, then we
obtain this value. Now, technically speaking, one is not in
the interval of convergence for this series, but hey,
trust me. You can trust me. This actually does converge. Now, if you believe that, then what
happens when you multiply everything by 1 half? Well of course, you just multiply each
term by 1 half. That is definitely true. Now, if we spread those terms out a little
bit and add everything together term wise.
What happens? Well, we get 1, negative 1 half plus 1
half is 0.
We pull down the 1 third. Negative 1 4th minus 1 4th, that's minus 1
half. Pull down the 1 5th.
The 1 6ths cancel. Pull down the 1 7th, etcetera. And I think you can keep going with this.
Some of the terms cancel. Some of the terms add together. What is this series.
well, it converges, and it converges as it must to 3 halves of 2.
That is, log 2 plus 1 half log 2. This is true.
Trust me. But at this point, we grow quite concern.
Because if we rearrange the terms, we get exactly what we started with.
It is a fact, that by rearranging the terms in the alternating harmonic series, we obtain a
different answer. That means your intuition is not so good
all the time. So, what, what can you trust? And you can't even trust me, sometimes I
make mistakes. There's one thing you can trust and that
is LOGIC. Careful deductive reasoning.
We re going to use tests for convergence or divergence
that are based on logic and calculus.
That is what you can trust. in general, the way that this is going to
to work is as follows. You'll start with some given sequence, and
you want to determine whether the series converges or diverges. You'll have several and about a half dozen
or more tests to choose from. Not all tests apply to all series. So you check to see whether a given test
applies. If it doesn't, choose another.
If it does, then apply that test. Some tests don't work with a given series, in which case you try again. But if the test does work, then you're
done. You've determined convergence or
divergence. Unfortunately, you might run out of tests,
in which case, you fail. But, that won't happen.
We're going to have lots of tests. Here's the first one.
This is called the nth term test, and it goes as follows.
It's an if then statement.
If the hypotheses are satisfied, then the conclusion follows.
Here's the hypothesis. If the limit as n goes to infinity of a
sub n is not equal to 0, then the series diverges. That's it.
Very, very simple. Now, in terms of applicability, this is a
four star test. It applies to any series. No problem.
It's easy to use. You just take the limit of the nth term as
n goes to infinity, and check whether it's zero or not.
But in terms of overall usefulness, well, this is very limited. In the sense that it only tells you one
thing, whether the series diverges. If that limit is equal to 0, you don't
know anything. What do I mean by that?
Let's look at some examples. If we consider the sum from 0 to infinity
of 4 to the n minus 3 to the n over 4 to the n plus 2
to the n.
Let's check it. The limit as n goes to infinity of the nth term is, after a little bit of factoring,
the limit as n goes to infinity of 1 minus 3
quarters to the n over 1 minus 1 half to the n. Well, 3 quarters to the n and 1 half to
the n both get very, very close to 0.
Therefore this limit is 1. Which is decidedly non zero. Therefore this series diverges according
to the nth term test. So far so good. What about the series then goes from zero to infinity negative 1 to the n over n
factorial. Well, if we take the limit of a sub n as n
goes to infinity, we get either plus or minus 1 in the numerator
and n factorial in the denominator. That is definitely zero. And indeed, this series converges. We know, that this is e to the x, 4x
equals negative 1. Simple enough. But what about the harmonic series, the
sum of 1 over n. What if we take the limit as n goes to
infinity, we get the limit of 1 over n, that's equal
to 0. But wait a minute, we already know that
this series diverges. What is wrong, has our test failed? No, our test is not failed.
Your logic has failed. If you have a logical statement of the form, if p then q, then you cannot conclude if q then p.
That is called the converse.
And the converse does not always hold. In our nth term test, we have the limit as
n goes to infinity of a sub n non zero, implies
divergence of the series. Divervegence of the series does not imply that the limit of the nth
term is non zero. The converse does not hold.
What then does hold? What can be said?
What can be said is the contrapositive. P implies q is the same thing as saying
that not q implies not p. We have to negate both statements and then reverse the arrow.
That contrapositive is always true.
So, what we can say is that, if the series converges, the negation of divergence,
then the limit, the nth term is zero. The contrapositive will hold. So, if you follow the rules of logic, you
won't get lost.
Or, if you do get lost, it's because you've not followed the rules of logic.
In our next two lessons, we'll begin introducing specific tests for measuring
convergence and divergence of infinite series.